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The Prisoners: A Logic Problem


Chaos Pudding

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Guest Chaos Pudding

100 prisoners are rounded up and made to stand in a big circle. The warden hands one prisoner a gun and tells him to shoot the person to his left, then pass the gun on to the next person (on the left). This continues until there is only one person left. If the first prisoner is number 1, which number would you have to be in order to survive? First to post the correct answer gets 1 Rep, I suppose.

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Guest Chaos Pudding

You guys aren't even trying. You disappoint me.

 

Oh, and to make it more interesting, I won't accept your answer unless you PM me your thought process. The first person to do that and get the right answer will get the Rep.

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Guest Chaos Pudding

Well I mean... If each shot kills one person then 1' date='3,5,7, etc... Would survive, unless you pass it back around after 100. I don't quit understand... =\ But I'm pretty sure 1 would be the last one standing.

[/quote']

 

What's not to understand? You're standing in a circle, you shoot the person standing to your left, and pass the gun to the next person on your left. You continue to do this until 1 person remains.

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Well I mean... If each shot kills one person then 1' date='3,5,7, etc... Would survive, unless you pass it back around after 100. I don't quit understand... =\ But I'm pretty sure 1 would be the last one standing.

[/quote']

 

What's not to understand? You're standing in a circle, you shoot the person standing to your left, and pass the gun to the next person on your left. You continue to do this until 1 person remains.

 

Oh I overlooked the circle part. >_<

 

I'll try and work this out.

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Prisoner # 73.

 

[spoiler=How I Got It]Luxord says:

1

5

9

13

17

21

25

29

33

37

41

45

49

53

57

61

65

69

73

77

81

85

89

93

97

99 is dead

97 kills him

And gives the gun to 1

BTM says:

Sh*t

I forgot 1

Luxord says:

HA I WASN'T THE ONLY ONE WHO MADE A MISTAKE

 

1

9

17

25

33

41

49

57

65

73

81

89

97

97 kills 1

gives the gun to 9

9

25

41

57

73

89

89 kills 97 gives the gun to 9

9

41

73

BTM says:

kk

you win

Luxord says:

73 kills 89 gives gun to 9

WOO.

 

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Guest Chaos Pudding

Note that, if you worked it out right, your process could apply to any number of prisoners (that are to the power of 10, I believe. Haven't tried it with a random large number). So you should be able to tell me who survives out of 10,000 prisoners, too.

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Prisoner #1 does the logical thing and shoots the warden; everybody lives except the warden.

 

More seriously:

 

After the first round, you live if your number is 1 mod 2.

After the second round, you live if your number is 1 mod 4.

After the third round, you live if your number is 5 mod 8.

After the fourth round, you live if your number is 5 mod 16.

After the fifth round, you live if your number is 5 mod 32.

Then 5 shoots 37.

Then 69 shoots 5.

Prisoner 69 wins.

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Prisoner #1 does the logical thing and shoots the warden; everybody lives except the warden.

 

More seriously:

 

After the first round' date=' you live if your number is 1 mod 2.

After the second round, you live if your number is 1 mod 4.

After the third round, you live if your number is 5 mod 8.

After the fourth round, you live if your number is 5 mod 16.

After the fifth round, you live if your number is 5 mod 32.

Then 5 shoots 37.

Then 69 shoots 5.

Prisoner 69 wins.

[/quote']

 

Prisoner 81 survives, though.

 

I solved it, using brute force.

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Prisoner #1 does the logical thing and shoots the warden; everybody lives except the warden.

 

More seriously:

 

After the first round' date=' you live if your number is 1 mod 2.

After the second round, you live if your number is 1 mod 4.

After the third round, you live if your number is 5 mod 8.

After the fourth round, you live if your number is 5 mod 16.

After the fifth round, you live if your number is 5 mod 32.

Then 5 shoots 37.

Then 69 shoots 5.

Prisoner 69 wins.

[/quote']

 

Prisoner 81 survives, though.

 

I solved it, using brute force.

 

...wait, you're right, I messed up. Round three ends with 1 mod 8 alive, not 5 mod 8.

 

IGNORE ME

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Prisoner #1 does the logical thing and shoots the warden; everybody lives except the warden.

 

More seriously:

 

After the first round' date=' you live if your number is 1 mod 2.

After the second round, you live if your number is 1 mod 4.

After the third round, you live if your number is 5 mod 8.

After the fourth round, you live if your number is 5 mod 16.

After the fifth round, you live if your number is 5 mod 32.

Then 5 shoots 37.

Then 69 shoots 5.

Prisoner 69 wins.

[/quote']

 

Prisoner 81 survives, though.

 

I solved it, using brute force.

 

Behindthemask is right I think. I forgot to remove prisoner 1 once, making my answer 79.

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[spoiler=I win]

In Round 1, the survivors are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, and 99, because every other prisoner is eliminated, leaving only the odd numbered prisoners. Therefore, we begin with 99 in Round 2.

 

In Round 2, the survivors are: 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, and 99. Again we begin with 99.

 

In Round 3, the survivors are: 7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87, and 95. Now we begin with 95. 99 is eliminated afterwards, which brings us to 7.

 

In Round 4, the survivors are: 7, 23, 39, 55, 71, and 87. 95 is eliminated, which brings us to 7 again.

 

In Round 5, the survivors are: 7, 39, and 71. 87 is eliminated, so we begin with 7 once more.

 

In Round 6, 7 kills 39 then passes the gun to 71, who then kills 7. Therefore, 71 is the winner.

 

 

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